Integrand size = 24, antiderivative size = 151 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}-\frac {a (12 b c-5 a d) \sqrt {c+d x^2}}{24 c^2 x^4}-\frac {\left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^3 x^2}+\frac {d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 91, 79, 44, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=\frac {d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{7/2}}-\frac {\sqrt {c+d x^2} \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right )}{16 c^3 x^2}-\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}-\frac {a \sqrt {c+d x^2} (12 b c-5 a d)}{24 c^2 x^4} \]
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Rule 44
Rule 65
Rule 79
Rule 91
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x^4 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} a (12 b c-5 a d)+3 b^2 c x}{x^3 \sqrt {c+d x}} \, dx,x,x^2\right )}{6 c} \\ & = -\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}-\frac {a (12 b c-5 a d) \sqrt {c+d x^2}}{24 c^2 x^4}+\frac {1}{16} \left (8 b^2-\frac {a d (12 b c-5 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}-\frac {a (12 b c-5 a d) \sqrt {c+d x^2}}{24 c^2 x^4}-\frac {\left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^3 x^2}+\frac {\left (d \left (-8 b^2+\frac {a d (12 b c-5 a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{32 c} \\ & = -\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}-\frac {a (12 b c-5 a d) \sqrt {c+d x^2}}{24 c^2 x^4}-\frac {\left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^3 x^2}+\frac {\left (-8 b^2+\frac {a d (12 b c-5 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{16 c} \\ & = -\frac {a^2 \sqrt {c+d x^2}}{6 c x^6}-\frac {a (12 b c-5 a d) \sqrt {c+d x^2}}{24 c^2 x^4}-\frac {\left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^3 x^2}+\frac {d \left (8 b^2-\frac {a d (12 b c-5 a d)}{c^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{3/2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2} \left (24 b^2 c^2 x^4+12 a b c x^2 \left (2 c-3 d x^2\right )+a^2 \left (8 c^2-10 c d x^2+15 d^2 x^4\right )\right )}{48 c^3 x^6}+\frac {d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{7/2}} \]
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Time = 2.94 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(-\frac {-\frac {15 \left (a^{2} d^{2}-\frac {12}{5} a b c d +\frac {8}{5} b^{2} c^{2}\right ) x^{6} d \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{8}+\left (\left (3 b^{2} x^{4}+3 a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}+\frac {15 x^{2} \left (\left (-\frac {12 b \,x^{2}}{5}-\frac {2 a}{3}\right ) c^{\frac {3}{2}}+a \sqrt {c}\, d \,x^{2}\right ) d a}{8}\right ) \sqrt {d \,x^{2}+c}}{6 c^{\frac {7}{2}} x^{6}}\) | \(117\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (15 a^{2} d^{2} x^{4}-36 x^{4} a b c d +24 b^{2} c^{2} x^{4}-10 a^{2} c d \,x^{2}+24 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right )}{48 c^{3} x^{6}}+\frac {\left (5 a^{2} d^{2}-12 a b c d +8 b^{2} c^{2}\right ) d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{16 c^{\frac {7}{2}}}\) | \(131\) |
default | \(a^{2} \left (-\frac {\sqrt {d \,x^{2}+c}}{6 c \,x^{6}}-\frac {5 d \left (-\frac {\sqrt {d \,x^{2}+c}}{4 c \,x^{4}}-\frac {3 d \left (-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{6 c}\right )+b^{2} \left (-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}\right )+2 a b \left (-\frac {\sqrt {d \,x^{2}+c}}{4 c \,x^{4}}-\frac {3 d \left (-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )\) | \(227\) |
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Time = 0.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=\left [\frac {3 \, {\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} \sqrt {c} x^{6} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (8 \, a^{2} c^{3} + 3 \, {\left (8 \, b^{2} c^{3} - 12 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{4} + 2 \, {\left (12 \, a b c^{3} - 5 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, c^{4} x^{6}}, -\frac {3 \, {\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (8 \, a^{2} c^{3} + 3 \, {\left (8 \, b^{2} c^{3} - 12 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{4} + 2 \, {\left (12 \, a b c^{3} - 5 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, c^{4} x^{6}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (146) = 292\).
Time = 50.93 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=- \frac {a^{2}}{6 \sqrt {d} x^{7} \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {a^{2} \sqrt {d}}{24 c x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {5 a^{2} d^{\frac {3}{2}}}{48 c^{2} x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {5 a^{2} d^{\frac {5}{2}}}{16 c^{3} x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {5 a^{2} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{16 c^{\frac {7}{2}}} - \frac {a b}{2 \sqrt {d} x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {a b \sqrt {d}}{4 c x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {3 a b d^{\frac {3}{2}}}{4 c^{2} x \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{4 c^{\frac {5}{2}}} - \frac {b^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 c x} + \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2 c^{\frac {3}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=\frac {b^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, c^{\frac {3}{2}}} - \frac {3 \, a b d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{4 \, c^{\frac {5}{2}}} + \frac {5 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{16 \, c^{\frac {7}{2}}} - \frac {\sqrt {d x^{2} + c} b^{2}}{2 \, c x^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a b d}{4 \, c^{2} x^{2}} - \frac {5 \, \sqrt {d x^{2} + c} a^{2} d^{2}}{16 \, c^{3} x^{2}} - \frac {\sqrt {d x^{2} + c} a b}{2 \, c x^{4}} + \frac {5 \, \sqrt {d x^{2} + c} a^{2} d}{24 \, c^{2} x^{4}} - \frac {\sqrt {d x^{2} + c} a^{2}}{6 \, c x^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=-\frac {\frac {3 \, {\left (8 \, b^{2} c^{2} d^{2} - 12 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{3}} + \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} - 48 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt {d x^{2} + c} b^{2} c^{4} d^{2} - 36 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d^{3} + 96 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{3} - 60 \, \sqrt {d x^{2} + c} a b c^{3} d^{3} + 15 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{4} - 40 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{4} + 33 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{4}}{c^{3} d^{3} x^{6}}}{48 \, d} \]
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Time = 5.96 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \sqrt {c+d x^2}} \, dx=\frac {\frac {{\left (d\,x^2+c\right )}^{5/2}\,\left (5\,a^2\,d^3-12\,a\,b\,c\,d^2+8\,b^2\,c^2\,d\right )}{16\,c^3}-\frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (5\,a^2\,d^3-12\,a\,b\,c\,d^2+6\,b^2\,c^2\,d\right )}{6\,c^2}+\frac {\sqrt {d\,x^2+c}\,\left (11\,a^2\,d^3-20\,a\,b\,c\,d^2+8\,b^2\,c^2\,d\right )}{16\,c}}{3\,c\,{\left (d\,x^2+c\right )}^2-3\,c^2\,\left (d\,x^2+c\right )-{\left (d\,x^2+c\right )}^3+c^3}+\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (5\,a^2\,d^2-12\,a\,b\,c\,d+8\,b^2\,c^2\right )}{16\,c^{7/2}} \]
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